I think this thread is kind of silly, but to play along, I just came up with a formula. I am not sure what the lowest averages for computer cubes are, so I not sure how values for
n greater than 7 compare with human records (computer or by hand).
The
Potential time (in seconds) for world record averages of 5 for a cube of size
n is:
\( P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)} \)
, where
\( {f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n. \)
\( {f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}} \)
\( \text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)} \)
Which I first showed in
this post .
\( \left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \) is power of ten of the rate of change for the number of positions for the nxnxn.
So for n = 1 to 1000 (ignore the value for n = 1), if we use the actual derivative (rate of change formula), with Mathematica input:
Spoiler
AccountingForm[(7.64/(3^2/Log[3]*
Yields (in seconds, but I included minutes, hours, days, and years)
Code:
n Seconds Minutes Hours Days Years
Code:
2 2.05022 0.034170333 0.00057 2.37294E-05 6.5012E-08
3 7.64 0.127333333 0.002122 8.84259E-05 2.42263E-07
4 24.0901 0.401501667 0.006692 0.000278821 7.63892E-07
5 52.427 0.873783333 0.014563 0.000606794 1.66245E-06
6 105.288 1.7548 0.029247 0.001218611 3.33866E-06
7 181.161 3.01935 0.050323 0.002096771 5.74458E-06
8 299.332 4.988866667 0.083148 0.003464491 9.49176E-06
9 456.763 7.612716667 0.126879 0.005286609 1.44839E-05
10 678.896 11.31493333 0.188582 0.007857593 2.15277E-05
11 956.883 15.94805 0.265801 0.011075035 3.03426E-05
12 1325.37 22.0895 0.368158 0.015339931 4.20272E-05
13 1773.2 29.55333333 0.492556 0.020523148 5.62278E-05
14 2341.76 39.02933333 0.650489 0.027103704 7.42567E-05
15 3010.87 50.18116667 0.836353 0.034848032 9.54741E-05
16 3833.91 63.8985 1.064975 0.044373958 0.000121572
17 4788.17 79.80283333 1.330047 0.055418634 0.000151832
18 5933.95 98.89916667 1.648319 0.068679977 0.000188164
19 7236.12 120.602 2.010033 0.083751389 0.000229456
20 8770.03 146.1671667 2.436119 0.101504977 0.000278096
21 10498.2 174.97 2.916167 0.121506944 0.000332896
22 12502.8 208.38 3.473 0.144708333 0.000396461
23 14730.2 245.5033333 4.091722 0.170488426 0.000467092
24 17281 288.0166667 4.800278 0.200011574 0.000547977
25 20099.9 334.9983333 5.583306 0.232637731 0.000637364
26 23293.6 388.2266667 6.470444 0.269601852 0.000738635
27 26786.9 446.4483333 7.440806 0.310033565 0.000849407
28 30708.6 511.81 8.530167 0.355423611 0.000973763
29 34982.6 583.0433333 9.717389 0.404891204 0.001109291
30 39743 662.3833333 11.03972 0.459988426 0.001260242
31 44889.9 748.165 12.46942 0.519559028 0.001423449
32 50596.2 843.27 14.0545 0.585604167 0.001604395
33 56736.8 945.6133333 15.76022 0.656675926 0.001799112
34 63473.7 1057.895 17.63158 0.734649306 0.002012738
35 70722.9 1178.715 19.64525 0.818552083 0.002242608
36 78634.4 1310.573333 21.84289 0.91012037 0.00249348
37 87097.2 1451.62 24.19367 1.008069444 0.002761834
38 96307.7 1605.128333 26.75214 1.114672454 0.003053897
39 106108 1768.466667 29.47444 1.228101852 0.003364663
40 116747 1945.783333 32.42972 1.351238426 0.003702023
41 128013 2133.55 35.55917 1.481631944 0.004059266
42 140196 2336.6 38.94333 1.622638889 0.004445586
43 153105 2551.75 42.52917 1.772048611 0.004854928
44 166969 2782.816667 46.38028 1.932511574 0.005294552
45 181623 3027.05 50.45083 2.102118056 0.005759228
46 197335 3288.916667 54.81528 2.283969907 0.006257452
47 213877 3564.616667 59.41028 2.475428241 0.006781995
48 231588 3859.8 64.33 2.680416667 0.007343607
49 250197 4169.95 69.49917 2.895798611 0.007933695
50 270037 4500.616667 75.01028 3.125428241 0.008562817
51 290842 4847.366667 80.78944 3.366226852 0.009222539
52 312969 5216.15 86.93583 3.622326389 0.009924182
53 336162 5602.7 93.37833 3.890763889 0.010659627
54 360772 6012.866667 100.2144 4.175601852 0.011440005
55 386523 6442.05 107.3675 4.473645833 0.012256564
56 413755 6895.916667 114.9319 4.788831019 0.013120085
57 442204 7370.066667 122.8344 5.118101852 0.014022197
58 472268 7871.133333 131.1856 5.466064815 0.01497552
59 503629 8393.816667 139.8969 5.829039352 0.015969971
60 536670 8944.5 149.075 6.211458333 0.017017694
61 571089 9518.15 158.6358 6.609826389 0.018109113
62 607331 10122.18333 168.7031 7.029293981 0.01925834
63 645037 10750.61667 179.1769 7.465706019 0.020453989
64 684631 11410.51667 190.1753 7.923969907 0.021709507
65 725775 12096.25 201.6042 8.400173611 0.023014174
66 768961 12816.01667 213.6003 8.900011574 0.024383593
67 813785 13563.08333 226.0514 9.41880787 0.025804953
68 860720 14345.33333 239.0889 9.962037037 0.027293252
69 909382 15156.36667 252.6061 10.52525463 0.028836314
70 960318 16005.3 266.755 11.11479167 0.030451484
71 1013076 16884.6 281.41 11.72541667 0.032124429
72 1068176 17802.93333 296.7156 12.36314815 0.033871639
73 1125192 18753.2 312.5533 13.02305556 0.035679604
74 1184782 19746.36667 329.1061 13.71275463 0.037569191
75 1246273 20771.21667 346.1869 14.42445602 0.039519058
76 1310464 21841.06667 364.0178 15.16740741 0.041554541
77 1376710 22945.16667 382.4194 15.93414352 0.043655188
78 1445727 24095.45 401.5908 16.73295139 0.045843702
79 1516896 25281.6 421.36 17.55666667 0.048100457
80 1591031 26517.18333 441.9531 18.41471065 0.050451262
81 1667420 27790.33333 463.1722 19.29884259 0.052873541
82 1746845 29114.08333 485.2347 20.21811343 0.055392092
83 1828696 30478.26667 507.9711 21.16546296 0.05798757
84 1913650 31894.16667 531.5694 22.14872685 0.060681443
85 2001209 33353.48333 555.8914 23.1621412 0.063457921
86 2092009 34866.81667 581.1136 24.21306713 0.06633717
87 2185378 36422.96667 607.0494 25.29372685 0.069297882
88 2282276 38037.93333 633.9656 26.41523148 0.072370497
89 2381854 39697.56667 661.6261 27.56775463 0.075528095
90 2485031 41417.18333 690.2864 28.76193287 0.078799816
91 2591079 43184.65 719.7442 29.98934028 0.082162576
92 2700875 45014.58333 750.2431 31.26012731 0.085644184
93 2813577 46892.95 781.5492 32.56454861 0.089217941
94 2930174 48836.23333 813.9372 33.91405093 0.092915208
95 3049880 50831.33333 847.1889 35.29953704 0.09671106
96 3173546 52892.43333 881.5406 36.73085648 0.100632484
97 3300440 55007.33333 916.7889 38.19953704 0.104656266
98 3431631 57193.85 953.2308 39.71795139 0.108816305
99 3565988 59433.13333 990.5522 41.27300926 0.113076738
100 3704803 61746.71667 1029.112 42.87966435 0.117478532
101 3846999 64116.65 1068.611 44.52545139 0.121987538
102 3993818 66563.63333 1109.394 46.22474537 0.126643138
103 4144043 69067.38333 1151.123 47.96346065 0.131406742
104 4299055 71650.91667 1194.182 49.75758102 0.13632214
105 4457701 74295.01667 1238.25 51.59376157 0.141352771
106 4621198 77019.96667 1283.666 53.48608796 0.146537227
107 4788563 79809.38333 1330.156 55.42318287 0.151844337
108 4960951 82682.51667 1378.042 57.41841435 0.157310724
109 5137229 85620.48333 1427.008 59.45866898 0.162900463
110 5318815 88646.91667 1477.449 61.5603588 0.168658517
111 5504309 91738.48333 1528.975 63.70728009 0.174540493
112 5695285 94921.41667 1582.024 65.91765046 0.180596303
113 5890419 98173.65 1636.228 68.17614583 0.186783961
114 6091219 101520.3167 1692.005 70.50021991 0.193151287
115 6296190 104936.5 1748.942 72.87256944 0.199650875
116 6507007 108450.1167 1807.502 75.31258102 0.206335838
117 6722257 112037.6167 1867.294 77.80390046 0.213161371
118 6943540 115725.6667 1928.761 80.3650463 0.220178209
119 7169398 119489.9667 1991.499 82.97914352 0.227340119
120 7401214 123353.5667 2055.893 85.66219907 0.234690956
121 7638011 127300.1833 2121.67 88.40290509 0.24219974
122 7880951 131349.1833 2189.153 91.21471065 0.249903317
123 8128887 135481.45 2258.024 94.08434028 0.257765316
124 8383292 139721.5333 2328.692 97.02884259 0.265832445
125 8642702 144045.0333 2400.751 100.0312731 0.274058283
126 8908916 148481.9333 2474.699 103.1124537 0.282499873
127 9180138 153002.3 2550.038 106.2515972 0.291100266
128 9458362 157639.3667 2627.323 109.4717824 0.299922692
129 9742041 162367.35 2706.123 112.7551042 0.308918094
130 10032622 167210.3667 2786.839 116.1183102 0.318132357
131 10328813 172146.8833 2869.115 119.5464468 0.327524512
132 10632258 177204.3 2953.405 123.0585417 0.337146689
133 10941313 182355.2167 3039.254 126.6355671 0.346946759
134 11257985 187633.0833 3127.218 130.3007523 0.356988363
135 11580260 193004.3333 3216.739 134.030787 0.367207636
136 11910360 198506 3308.433 137.8513889 0.377675038
137 12246555 204109.25 3401.821 141.7425347 0.388335712
138 12590449 209840.8167 3497.347 145.7227894 0.399240519
139 12940600 215676.6667 3594.611 149.775463 0.410343734
140 13298831 221647.1833 3694.12 153.9216551 0.421703165
141 13663309 227721.8167 3795.364 158.1401505 0.433260686
142 14036259 233937.65 3898.961 162.4567014 0.445086853
143 14415622 240260.3667 4004.339 166.8474769 0.457116375
144 14803493 246724.8833 4112.081 171.3367245 0.469415684
145 15197942 253299.0333 4221.651 175.9021065 0.481923579
146 15601305 260021.75 4333.696 180.5706597 0.494714136
147 16011416 266856.9333 4447.616 185.3173148 0.507718671
148 16430474 273841.2333 4564.021 190.1675231 0.521006913
149 16856448 280940.8 4682.347 195.0977778 0.53451446
150 17291789 288196.4833 4803.275 200.1364468 0.548319032
151 17734220 295570.3333 4926.172 205.2571759 0.562348427
152 18186046 303100.7667 5051.679 210.4866435 0.576675736
153 18645135 310752.25 5179.204 215.8001736 0.591233352
154 19114053 318567.55 5309.459 221.2274653 0.606102645
155 19590412 326506.8667 5441.781 226.7408796 0.621207889
156 20076625 334610.4167 5576.84 232.3683449 0.636625602
157 20570454 342840.9 5714.015 238.0839583 0.652284817
158 21074587 351243.1167 5854.052 243.918831 0.66827077
159 21586517 359775.2833 5996.255 249.8439468 0.684503964
160 22108771 368479.5167 6141.325 255.8885532 0.701064529
161 22639228 377320.4667 6288.674 262.0281019 0.717885211
162 23180021 386333.6833 6438.895 268.2872801 0.735033644
163 23729202 395486.7 6591.445 274.6435417 0.752448059
164 24289187 404819.7833 6746.996 281.1248495 0.770205067
165 24857754 414295.9 6904.932 287.7054861 0.788234209
166 25437129 423952.15 7065.869 294.4112153 0.806606069
167 26025278 433754.6333 7229.244 301.2184954 0.825256152
168 26624963 443749.3833 7395.823 308.159294 0.844272038
169 27233128 453885.4667 7564.758 315.1982407 0.863556824
170 27853080 464218 7736.967 322.3736111 0.883215373
171 28481949 474699.15 7911.653 329.6521875 0.903156678
172 29122608 485376.8 8089.613 337.0672222 0.923471842
173 29772633 496210.55 8270.176 344.5906597 0.944083999
174 30434443 507240.7167 8454.012 352.2504977 0.965069857
175 31105816 518430.2667 8640.504 360.0210185 0.986358955
176 31789754 529829.2333 8830.487 367.9369676 1.008046487
177 32482929 541382.15 9023.036 375.9598264 1.030026922
178 33188929 553148.8167 9219.147 384.1311227 1.052414035
179 33904631 565077.1833 9417.953 392.4147106 1.075108796
180 34633150 577219.1667 9620.319 400.8466435 1.098209982
181 35371849 589530.8167 9825.514 409.3964005 1.121633974
182 36123631 602060.5167 10034.34 418.097581 1.145472825
183 36885519 614758.65 10245.98 426.9157292 1.169632135
184 37660753 627679.2167 10461.32 435.8883449 1.194214644
185 38446584 640776.4 10679.61 444.9836111 1.219133181
186 39245739 654095.65 10901.59 454.2330903 1.24447422
187 40055702 667595.0333 11126.58 463.607662 1.270157978
188 40879849 681330.8167 11355.51 473.1464005 1.296291508
189 41714421 695240.35 11587.34 482.8057986 1.322755613
190 42563453 709390.8833 11823.18 492.6325579 1.349678241
191 43423421 723723.6833 12062.06 502.5858912 1.376947647
192 44298133 738302.2167 12305.04 512.7098727 1.404684583
193 45183681 753061.35 12551.02 522.9592708 1.432765126
194 46084254 768070.9 12801.18 533.3825694 1.461322108
195 46996189 783269.8167 13054.5 543.9373727 1.490239377
196 47923439 798723.9833 13312.07 554.6694329 1.519642282
197 48861942 814365.7 13572.76 565.5317361 1.549402017
198 49816046 830267.4333 13837.79 576.5746065 1.579656456
199 50781945 846365.75 14106.1 587.7539931 1.610284912
200 51763739 862728.9833 14378.82 599.1173495 1.641417396
201 52757212 879286.8667 14654.78 610.6158796 1.672920218
202 53766871 896114.5167 14935.24 622.3017477 1.704936295
203 54788766 913146.1 15219.1 634.1292361 1.737340373
204 55827146 930452.4333 15507.54 646.1475231 1.770267187
205 56877989 947966.4833 15799.44 658.3100579 1.8035892
206 57944910 965748.5 16095.81 670.6586806 1.837421043
207 59025225 983753.75 16395.9 683.1623264 1.871677607
208 60121910 1002031.833 16700.53 695.8554398 1.90645326
209 61231869 1020531.15 17008.85 708.7021875 1.941649829
210 62358851 1039314.183 17321.9 721.7459606 1.977386194
211 63498976 1058316.267 17638.6 734.9418519 2.01353932
212 64656422 1077607.033 17960.12 748.3382176 2.050241692
213 65827614 1097126.9 18285.45 761.8936806 2.087379947
214 67016432 1116940.533 18615.68 775.6531481 2.125077118
215 68218856 1136980.933 18949.68 789.5700926 2.163205733
216 69439593 1157326.55 19288.78 803.6989931 2.201915049
217 70674175 1177902.917 19631.72 817.9881366 2.241063388
218 71926994 1198783.233 19979.72 832.4883565 2.280790018
219 73193892 1219898.2 20331.64 847.1515278 2.32096309
220 74479332 1241322.2 20688.7 862.0293056 2.361724125
221 75779488 1262991.467 21049.86 877.0774074 2.402951801
222 77098502 1284975.033 21416.25 892.3437731 2.444777461
223 78432482 1307208.033 21786.8 907.7833565 2.487077689
224 79785225 1329753.75 22162.56 923.4401042 2.529972888
225 81153180 1352553 22542.55 939.2729167 2.573350457
226 82540623 1375677.05 22927.95 955.3312847 2.617345986
227 83943110 1399051.833 23317.53 971.5637731 2.661818557
228 85365827 1422763.783 23712.73 988.0304051 2.706932617
229 86803836 1446730.6 24112.18 1004.674028 2.752531583
230 88261975 1471032.917 24517.22 1021.550637 2.798768867
231 89735650 1495594.167 24926.57 1038.60706 2.845498795
232 91230214 1520503.567 25341.73 1055.905255 2.892891109
233 92740568 1545676.133 25761.27 1073.386204 2.94078412
234 94271702 1571195.033 26186.58 1091.107662 2.98933606
235 95818873 1596981.217 26616.35 1109.014734 3.038396531
236 97387601 1623126.683 27052.11 1127.171308 3.08814057
237 98972168 1649536.133 27492.27 1145.511204 3.138386859
238 100578626 1676310.433 27938.51 1164.104468 3.189327308
239 102202094 1703368.233 28389.47 1182.894606 3.240807141
240 103847337 1730788.95 28846.48 1201.936771 3.292977454
241 105508917 1758481.95 29308.03 1221.168021 3.345665811
242 107193544 1786559.067 29775.98 1240.666019 3.399084982
243 108894764 1814912.733 30248.55 1260.356065 3.453030315
244 110618907 1843648.45 30727.47 1280.311424 3.50770253
245 112360376 1872672.933 31211.22 1300.467315 3.56292415
246 114124631 1902077.183 31701.29 1320.886933 3.618868309
247 115906471 1931774.517 32196.24 1341.510081 3.675370085
248 117711932 1961865.533 32697.76 1362.40662 3.732620878
249 119535246 1992254.1 33204.24 1383.509792 3.790437785
250 121382035 2023033.917 33717.23 1404.884664 3.84899908
251 123246938 2054115.633 34235.26 1426.46919 3.908134767
252 125136170 2085602.833 34760.05 1448.335301 3.96804192
253 127043789 2117396.483 35289.94 1470.414225 4.028532122
254 128975579 2149592.983 35826.55 1492.772905 4.089788781
255 130926544 2182109.067 36368.48 1515.353519 4.151653475
256 132901511 2215025.183 36917.09 1538.211933 4.214279268
257 134895928 2248265.467 37471.09 1561.295463 4.277521816
258 136915756 2281929.267 38032.15 1584.673102 4.341570142
259 138954253 2315904.217 38598.4 1608.266817 4.406210458
260 141018522 2350308.7 39171.81 1632.158819 4.471667998
261 143102268 2385037.8 39750.63 1656.27625 4.537743151
262 145211609 2420193.483 40336.56 1680.689919 4.604629915
263 147341251 2455687.517 40928.13 1705.338553 4.67216042
264 149496299 2491604.983 41526.75 1730.281238 4.740496544
265 151671931 2527865.517 42131.09 1755.462164 4.809485382
266 153874446 2564574.1 42742.9 1780.954236 4.879326674
267 156096713 2601611.883 43360.2 1806.674919 4.949794299
268 158346232 2639103.867 43985.06 1832.711019 5.021126078
269 160616349 2676939.15 44615.65 1858.985521 5.093111016
270 162913520 2715225.333 45253.76 1885.573148 5.165953831
271 165232151 2753869.183 45897.82 1912.409155 5.239477137
272 167578212 2792970.2 46549.5 1939.562639 5.313870244
273 169945441 2832424.017 47207.07 1966.961123 5.388934583
274 172340469 2872341.15 47872.35 1994.681354 5.464880422
275 174757546 2912625.767 48543.76 2022.656782 5.541525431
276 177202802 2953380.033 49223 2050.958356 5.61906399
277 179669804 2994496.733 49908.28 2079.51162 5.697292111
278 182165359 3036089.317 50601.49 2108.395359 5.776425641
279 184683560 3078059.333 51300.99 2137.541204 5.85627727
280 187230081 3120501.35 52008.36 2167.014826 5.937026922
281 189800169 3163336.15 52722.27 2196.761215 6.018523877
282 192398954 3206649.233 53444.15 2226.839745 6.100930809
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882 15350173210 255836220.2 4263937 177664.0418 486.7507994
883 15417340351 256955672.5 4282595 178441.4392 488.8806555
884 15484747022 258079117 4301319 179221.6091 491.018107
885 15552342559 259205709.3 4320095 180003.9648 493.1615474
886 15620188880 260336481.3 4338941 180789.2231 495.3129401
887 15688224878 261470414.6 4357840 181576.6768 497.4703475
888 15756512726 262608545.4 4376809 182367.0454 499.6357409
889 15824985893 263749764.9 4395829 183159.5589 501.8070108
890 15893711961 264895199.4 4414920 183954.9995 503.9863001
891 15962629306 266043821.8 4434064 184752.654 506.1716548
892 16031800624 267196677.1 4453278 185553.248 508.3650629
893 16101158823 268352647.1 4472544 186356.0049 510.564397
894 16170772047 269512867.5 4491881 187161.7135 512.7718178
895 16240578157 270676302.6 4511272 187969.6546 514.9853551
896 16310640369 271844006.2 4530733 188780.5598 517.2070132
897 16380891030 273014850.5 4550248 189593.6462 519.4346471
898 16451398850 274189980.8 4569833 190409.7089 521.6704354
899 16522101171 275368352.9 4589473 191228.0228 523.9123913
900 16593061731 276551028.9 4609184 192049.3256 526.1625359
901 16664212314 277736871.9 4628948 192872.8277 528.4187061
902 16735622198 278927036.6 4648784 193699.331 530.6830986
903 16807228204 280120470.1 4668675 194528.1042 532.9537102
904 16879094595 281318243.3 4688637 195359.8911 535.2325785
905 16951157933 282519298.9 4708655 196193.9576 537.5176919
906 17023472036 283724533.9 4728742 197030.9263 539.8107571
907 17095994596 284933243.3 4748887 197870.3078 542.1104324
908 17168768965 286146149.4 4769102 198712.6038 544.4180925
909 17241747277 287362454.6 4789374 199557.2602 546.7322196
910 17314983821 288583063.7 4809718 200404.9053 549.0545352
911 17388419750 289806995.8 4830117 201254.8582 551.3831732
912 17462114960 291035249.3 4850587 202107.812 553.720033
913 17536015793 292266929.9 4871115 202963.1458 556.063413
914 17610176982 293502949.7 4891716 203821.4928 558.4150489
915 17684539194 294742319.9 4912372 204682.1666 560.7730592
916 17759168279 295986138 4933102 205545.9292 563.1395319
917 17833999221 297233320.4 4953889 206412.028 565.5124055
918 17909092649 298484877.5 4974748 207281.1649 567.8936025
919 17984388748 299739812.5 4995664 208152.6475 570.2812262
920 18059948390 300999139.8 5016652 209027.1804 572.6772067
921 18135717037 302261950.6 5037699 209904.1324 575.0798147
922 18211750308 303529171.8 5058820 210784.1471 577.4908139
923 18287993443 304799890.7 5079998 211666.5908 579.9084679
924 18364496735 306074945.6 5101249 212552.0455 582.3343714
925 18441210731 307353512.2 5122559 213439.939 584.7669562
926 18518191498 308636525 5143942 214330.9201 587.2080003
927 18595378248 309922970.8 5165383 215224.2853 589.6555761
928 18672838433 311213973.9 5186900 216120.8152 592.1118225
929 18750505442 312508424 5208474 217019.7389 594.5746272
930 18828441379 313807356.3 5230123 217921.7752 597.0459595
931 18906584963 315109749.4 5251829 218826.2148 599.5238763
932 18985004185 316416736.4 5273612 219733.8447 602.0105335
933 19063631902 317727198.4 5295453 220643.8878 604.5038021
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935 19221638834 320360647.2 5339344 222472.6717 609.514169
936 19301024810 321683746.8 5361396 223391.4909 612.0314818
937 19380615256 323010254.3 5383504 224312.6766 614.5552783
938 19460490361 324341506 5405692 225237.157 617.0881012
939 19540570764 325676179.4 5427936 226164.0135 619.6274342
940 19620931239 327015520.7 5450259 227094.1116 622.1756481
941 19701497819 328358297 5472638 228026.5951 624.7303976
942 19782351328 329705855.5 5495098 228962.3996 627.2942456
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951 20520945695 342015761.6 5700263 237510.9455 650.7149193
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955 20855707126 347595118.8 5793252 241385.4991 661.3301346
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970 22147254634 369120910.6 6152015 256333.9657 702.2848375
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978 22859854725 380997578.8 6349960 264581.6519 724.8812381
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994 24335926067 405598767.8 6759979 281665.811 771.6871533
995 24430453352 407174222.5 6786237 282759.8768 774.6845939
996 24525286307 408754771.8 6812580 283857.4804 777.6917271
997 24620355305 410339255.1 6838988 284957.816 780.7063453
998 24715737515 411928958.6 6865483 286061.7768 783.7308953
999 24811356666 413522611.1 6892044 287168.4799 786.7629587
1000 24907283784 415121396.4 6918690 288278.7475 789.8047877
For \( \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor \), I made the approximation formula (which is accurate up to n = 31).
\( \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor \approx \frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right) \)
which I first showed in
this post .
By using this approximation, we can have
a formula at wolframalpha.
\( \frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left( 21 \right)}\frac{n^{2}}{\ln \left( n \right)}\frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right) \)
Note that \( \left\lfloor \log _{10}\left( {f}'\left( 3 \right) \right) \right\rfloor =21 \) with the approximation formula as well as the actual derivative formula (that's where the 21 comes from).
To get values for a certain
n , just change the 2 (located on the far right of the input bar) to a different number in the input bar.
